Is there a way that when something is subdivided, it can be constrained from making the zig-zag flower petal pointed edges?
While I mostly agree with the answer offered by cegaton, in my opinion, it has one characteristic that makes it undesirable in some circumstances. That characteristic is that the center point has a number of duplicate vertices in the same location. In this class of cases, I have found a useful solution in turning the triangle fan into a quadrangle fan.
Starting with cegaton's first illustration, instead of deleting the center vertex, (as at 'A' below), instead deselect everything, switch to edge select mode, and select every other edge between the center vertex and the surrounding edge loop, (as at 'B' below), and delete those edges. Then create quadrangles with the remaining edges, with the result as in 'C' below.
This achieves a similar appearance to the option provided by cegaton, with the exception that it has only a single center vertex.
One warning: this works best with meshes which have an even number of vertices in the circumference.
The problem is that you have a "triangle fan": A bunch of triangles converging on a single vertex (also known as a pole). The subsurf modifier will not work well with that.
You need quads.
Delete the vertex at the center of all the triangles.
Select the inner ring loop and press E to extrude and without moving the mouse press escape
Then scale all of the vertices to 0 (S0)
That will make all of the newly created vertices converge in a single place. And the subsurf should work well.